Averages

What does the word average signify?  Did you know Wikipedia lists at least 14 different kinds of averages? (http://en.wikipedia.org/wiki/Average)

The most common average is the arithmetic mean, but it is not always appropriate.  Basil runs 800 metres at 8 metres per second and another 800 m at 7 metres per second.  What is his average speed?  Many ‘good’ students blithely add 8 and 7 and then divide by 2 to get 7.5. wrong!

Let’s take an extreme case.  Extreme cases are an essential part of your mathematical toolbox. Suppose Basil runs his first 800 m at 8 m/s and his second 800 m at the glacially slow speed of 0. 0000…1 m/s.  (Imagine as many zeros as you like before the one). Is the average speed (8 + 0)/2 = 4 metres per second?  Certainly not!

Imagine another case.  I am driving my car at 100 km per hour for 0.99999 …9 hours (or 59.999…9 minutes) and for a brief fleeting instant at 200 km per h.  Is my average speed (100+200)/2 = 150 km/h?  Certainly not!

Instead of the ordinary mean (a + b)/2, what the student needs is a weighted mean (w₁ × a + w₂ × b)/( w₁ + w₂).   But, what are the appropriate weights?  The obvious (and correct answer) are the times spent at each speed.  Then we get (t₁ × s₁ + t₂ × s₂)/( t₁ + t₂) = (d₁ + d₂)/( t₁ + t₂) = total distance/total time, as taught by the physics teachers.